Time to event, e.g. disease specific mortality, is the primary endpoint in many clinical trials. The use of group
sequential boundaries in monitoring the trial is not only commonplace, but ethically mandated in all trials of human
subjects. The logrank statistic is often the monitoring statistic of choice due to its natural connection with the
relative risk, which is often the parameter of inference. This natural connection, which is based upon the assumption
of proportional hazards, admits a one-to-one correspondence between the inferential procedure based upon the usual
standard normal scale and that based on the scale of the natural parameter. However, the assumption of proportional
hazards is not always a reasonable assumption. In many subject areas, e.g. in disease-prevention trials, one expects
that the hazard ratio will not be constant. Much of the prior work on the use of the weighted logrank statistic in a
sequential design is confined to the use a weighting function from the \(G^{\rho , \gamma }(t) = S^{\rho }(t)(1-S(t))^{\gamma }\)  family, of Fleming and Harrington, [1]}. They suggest two major types of problems
which can arise. First, they argue that use of the weighted logrank statistic does not reproduce the single point
analysis in the way that is desired. Most notably, they argue, there is no clinically meaningful parameter that allows
the values of the monitoring statistic and sequential boundaries to be cast into a clinically meaningful scale. They
believe that this problem is further aggrivated when the range of the weighting function over the duration of the trial
is quite large, such as is the case with the \(G^{0, 1}\)  weight function (Gillen and Emerson, [2]}) and
suggest a re-weighting scheme whereby the most weight is given to the most recent data collected at each
analysis. Secondly, they argue that if the chosen weighting function is non-deterministic or trial-specific then it is
impossible to compare results from different clinical trials, (Gillen and Emerson, [3]}, [4]}). While the bulk of these cautious remarks are useful to know in their own right, several important
points have been omitted from the discussion. Firstly, as we will show, there is a natural, clinically meaningful
parameter, the weighted average logged relative risk, that is connected bijectively to the weighted logrank statistic
when there is no change in sign in the instantaneous logged relative risk. Under suitable shape assumptions, the
bijection can be estimated at each analysis. We will show that the asymptotic distribution of the WLR statistic,
suitably normalized is a Brownian motion plus drift under nothing but boundeness conditions. In two corollaries, we
demonstrate how each of two presented shape assumptions translates into a form of the drift function and consequently,
into an estimator of the weighted average logged relative risk. We then demonstrate how the usual results concerning
monitoring and end of trial estimation follow. Finally, we note that this bijection between the weighted logrank
statistic and the weighted average logged relative risk allows the values of the monitoring statistic, efficacy and
futility boundaries, and reported point estimate and confidence interval to be cast into a clinically meaningful scale.
